Visual Thinking in Mathematics

by

Jessica Carter

• LAST MODIFIED: 28 August 2019
• DOI: 10.1093/obo/9780195396577-0229

Introduction

In contemporary philosophy, “visual thinking in mathematics” refers to studies of the kinds and roles of visual representations in mathematics. Visual representations include both external representations (i.e., diagrams) and mental visualization. Currently, three main areas and questions are being investigated. The first concerns the roles of diagrams, or the diagram-based reasoning, found in Euclid’s Elements. Second is the epistemic role of diagrams: the question of whether reasoning based on diagrams can be rigorous. This debate includes the question of whether beliefs based on visual input can be justified, and whether visual perception may lead to mathematical knowledge. The third observes that diagrams abound in (contemporary) mathematical practice, and so tries to understand the role they play, going beyond the traditional debates on the legitimacy of using diagrams in mathematical proofs. Looking at the history of mathematics, one will find that it is only recently that diagrammatic proofs have become discredited. For about 2,000 years, Euclid’s Elements was conceived as the paradigm of (mathematical) rigorous reasoning, and so until the 18th century, Euclidean geometry served as the foundation of many areas of mathematics. One includes the early history of analysis, where the study of curves draws on results from (Euclidean) geometry. During the 18th and 19th centuries, however, diagrams gradually disappear from mathematical texts, and around 1900 one finds the famous statements of Pasch and Hilbert claiming that proofs must not rely on figures. The development of formal logic during the 20th century further contributed to a general acceptance of a view that the only value of figures, or diagrams, is heuristic, and that they have no place in mathematical rigorous proofs. A proof, according to this view, consists of a discrete sequence of sentences and is a symbolic object. In the latter half of the 20th century, philosophers, sensitive to the practice of mathematics, started to object to this view, leading to the emergence of the study of visual thinking in mathematics.

General Overviews

So far there are only a few general overviews. Each of these has a different focus, which means that together they provide a good overview of the field. In the first, Mancosu 2005, one finds a historical sketch of the reasons for abandoning diagrams in mathematical reasoning during the 19th century. Mancosu further explains how the invention of visualization techniques in computer science has renewed the interest of visualization in some areas of contemporary mathematics. In addition, there are two (as of 2018) recent surveys. One is Giaquinto 2015, a Stanford Encyclopedia entry on the epistemology of visual reasoning in mathematics. This entry gives a thorough introduction to some of the main questions in the area, including discussions concerning the cognitive and epistemic role of diagrams and the role of diagrams in contemporary mathematics. The second, Giardino 2017, presents central contributions to diagrammatic reasoning in Euclid, the work on the role of diagrams in contemporary algebra, topology, and analysis. In addition, the topics of visualization and automated systems of Euclidean geometry and number theory are mentioned. The final survey, Shin, et al. 2013, on diagrams in general, has a section on diagrams in Euclidean geometry.

• Giaquinto, Marcus. “The Epistemology of Visual Thinking in Mathematics.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford University, 2015.

  Contains a rich discussion of issues related to various roles of visual thinking in mathematics, including the epistemic role of diagrams, and their role for discovery and understanding. Includes a number of examples of diagram use from different parts of mathematics, such as representations of knots, the intermediate value theorem, contemporary algebra, and analysis.

• Giardino, Valeria. “Diagrammatic Reasoning in Mathematics.” In Springer Handbook on Model-Based Science. Edited by Lorenzo Magnani and Tommaso Bertolotti, 499–522. Springer Handbooks. Cham, Switzerland: Springer, 2017.

  DOI: 10.1007/978-3-319-30526-4_22

  Associates the renewed interest in diagrams with the emergence of the philosophy of mathematical practice. The main contributions on Euclidean diagrammatic reasoning are presented, including the developed formal systems of Euclidean reasoning. Gives an account of the role of diagrams in contemporary mathematics. It concludes with a section on visual thinking.

• Mancosu, Paolo. “Visualization in Logic and Mathematics.” In Visualization, Explanation and Reasoning Styles in Mathematics. Edited by Paolo Mancosu, Klaus F. Jørgensen, and Stig A. Pedersen, 13–30. Dordrecht, The Netherlands: Springer, 2005.

  DOI: 10.1007/1-4020-3335-4_2

  Contains a historical account on why visualization became discredited in the 19th century, mentioning the reservations of Pasch, Hilbert, and Russell. Also points to the discovery of nowhere differentiable continuous functions by Weierstrass and Koch. Considers the development of visualization techniques in computer science as a reason for the return of visualization as a heuristic tool in mathematics.

• Shin, Sun-Joo, Oliver Lemon, and John Mumma. “Diagrams.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford University, 2013.

  Covers work on diagrammatic reasoning in Euclid’s Elements.